In the process
I am in the process of doing this now. Not sure what I am going to go back with, but will not be vinyl. I hate that stuff. As for yours. Depending on the back, the best thing I would think is contact cement. If it has the foam back, the contact adhesive will probably destroy the backing. You might consider a tile adhesive, called clear set or thin set. You put this on the ceiling, let it set up, and then apply the vinyl. This I am pretty sure will stick to the vinyl, but not 100% sure it would stick to the foam backing.